“You know of course that a mathematical line, a line of thickness nil, has no real existence. They taught you that? Neither has a mathematical plane. These things are mere abstractions.”

“That’s all right,” said the Psychologist.

“Nor having only length, breadth, and thickness, can a cube have a real existence.”

“There I object,” said Filby. “Of course a solid body may exist. All real things —”

“So most people think. But wait a moment. Can an instantaneous cube exist?”

“Don’t follow,” said Filby.

Can a cube that does not last for any time at all, have a real existence?”

Filby became pensive.

“Clearly,” the Time Traveller proceeded, “any real body must have extension in four directions, it must have Length, Breadth, Thickness, and —Duration.”

The dialogue points to what is, in my experience, a much overlooked idea: that there is an interesting constraint applied to time by the first three spatial dimensions. When we look around, we don’t see triangles, we see things that look like triangles. This is the sort of thinking that led Plato to the idea of universal forms and the allegory of the Cave. The dialogue points out an interesting question: Supposing that one can obtain, say, a platonic solid, what if it exists only for an instant —that is, no duration at all? I don’t see this question come up often in the more academic forums; maybe it does and I am just missing it. The notion that time is infinitely divisible is one that has been around for a while, but my interest was piqued when I read that John Baez said, in a 1999 article, “Is time quantized? In other words, is there a fundamental unit of time that could not be divided into a briefer unit?”: “The brief answer to this question is, ‘Nobody knows.’ Certainly there is no experimental evidence in favor of such a minimal unit. On the other hand, there is no evidence against it, except that we have not yet found it.” William G. Unruh and Willaim G. Tifft assented to this, saying that there is no “direct” or “empirical” evidence to support the claim that space has a minimum length. Based what the fictitious scientist had to say, does anyone feel bothered by what the non-fictitious scientists had to say?

Granted, the issue is not an easy one to manage. With respect to the spatial dimension, debate has been going on, probably, since the first conception of distance measurement. One need simply to take a look at Immanuel Kant’s vacillating (well, he at least changed his mind on the matter officially once, but I suspect there was much indecision throughout his life) between whether space was infinitely divisible. I highly recommend Michael Friedman’s discussion on this topic, which can be found in the introduction to Kant’s Metaphysical Foundations of Natural Science (Cambridge Texts in the History of Science, Cambridge University Texts, 2004). Even today, many scholars debate the issue, such as Amit Hagar. In my view, time quantization seems to be a little bit more tractable than the spatial dimensions debate. In fact, the problem for space, which time doesn’t quite seem to suffer, is that of figuring up how in the world space could be quantized, at all. For example, do you chop space up into cubes? But then the minimum distance, along the edges, is undermined by the fact that the diagonal of the cube is not a multiple of the quantum length. With spheres, there will be an optimal configuration to minimize the “space” between the spheres, if that is how space is partitioned. But what is that “space” between spaces? Also, if the diameter is the quantized length, then there is the problem of the circumference not being a whole number of radius lengths in its length. Mathematically, it becomes even more of a mess when you add physics. Time is different. In the present, time can still easily be thought of as ontologically distinct from the spatial dimensions. After all, most people don’t confuse spatial and temporal coordinates.

The thing that is particularly stark in contrast, when the Time Traveller and the physicists’ thinking are juxtaposed, is the approach. It seems, really, quite difficult to argue against the platonic line of argument that no triangle exists in our world. Whether there is some magical place where the forms exist is something for the pure philosophers to argue about. It certainly isn’t something a philosopher of physics cares much about. So this much is certain enough: objects with no thickness don’t seem to exist in the universe. That’s not a big revelation, as most high school geometry students would tend to agree with this thought. What about the proposition of the Time Traveller? Does the Time Traveller’s idea about objects of thickness requiring non-zero extension in time, in order to exist? The proposal definitely appears to be a strong line of reasoning. After all, what does it mean for something to exist for no time? Again, I would leave that to the pure philosophers, because the only “thing” with spatial extension that I can think of as existing, which requires zero duration, is possibly the platonic forms.

Before continuing, please note that the line of reasoning does not work in the reverse, on behalf of platonic forms. One cannot wish to succeed in convincing anyone by saying, “Well, if time must have, at the very least, an absolute smallest fundamental length, then why wouldn’t space be similarly partitioned?” First, we did note the seeming distinction between the ontology of space and time, and that is granted the fact that we acknowledge that there is some relationship between space and time, as explicated by relativity. (I don’t think interrelation between space and time should cause conflation in regard to ontology.) Second, having extension in a body is not the same thing, exactly, as extension in space. This may seem problematic, at first, based on the preceding, but it is not. Those platonic forms we talked about —those are not necessarily objects, in the sense of everyday material objects; they may be thought of as spaces just as well. The point is that bodies definitely require extension to exist, and it may have some real, fundamental partitioning, but that doesn’t require the space to. In fact, I have some curiosities of my own regarding in what sense space could be divisible if substantivalism is not to hold —how radically anti-Kantian of me to suggest such! To summarize, bodies require extension, that is, space, and bodies in space require a non-zero time to be objects in our experiential sense, making it requisite that time not be infinitely divisible; but the body’s divisibility, being that it is composed of something, does not constrain the space in anyway (probably). The tricky part is that, in making the extension “real,” we assumed a mereology of substance.

This puts us where we want to be, in discussing objects of experience, as captured in the dialogue. Real objects necessitate duration of some kind, some non-zero extension of the time dimension. In my judgment, I think this precludes any notion that time is infinitely divisible. We are left with two obvious and interesting options. The first is that time in one uninterrupted sort of continuum, where the events sort of smear into one another. This idea, called “duration” and developed by Henri Bergson (see: Matter and Memory, Duration and Simultaneity, and Creative Evolution), and the view makes a great deal of sense out of our personal experience. For example, his idea of heterogeneous multiplicity is one of most original ideas every to come out of philosophy, and it is basically a consequence of Bergson’s idea of duration. However, there is still room for quantization, as well. I must point out, though, that the partitioning would have to be done such that each block of time is a mini-duration, in the Bergsonian vein. I can think of one alternative to this, which would be based on some much more complicated line of thought, supposing that space is quantized, but I leave that for later. I will say that, with regard to this complicated line of thought, where quantization of space and time is interrelated, this line of thought is the only one I have come up with that permits time to be instantaneous changes such that time is, both, not infinitely divisible and partitioned in Bergsonian durations.

Based on all of this discussion, I think the physicists were a bit too reticent in their thinking; there certainly are philosophical avenues by which one can say something constructive and near enough to matter of fact. I think what lies outside the realm of possibility is time’s infinite divisibility. That certainly has import to how we see the world mathematically, as Bergson explained. At least, I think, this one possibility can be ruled out, even in the physicist’s mind, so I would have like to have heard from the physicists something other than platitudinous statements of humility, like “experiment has told us nothing on the issue, and theory might suggest something.” While we have not concluded anything for or against quantization, I think we can say that Bergson’s relevance to the topic has increased.